Charu C. Aggarwal T J Watson Research Center IBM Corporation Hawthorne, NY USA On Biased Reservoir Sampling in the Presence of Stream Evolution VLDB Conference, Seoul, South Korea, 2006
Synopsis Construction in Data Streams • Synopsis maintenance is an important problem in massive volume applications such as data streams. • Many synopsis methods such as wavelets, histograms and sketches are designed for use with specific applications such as approximate query answering. • An important class of stream synopsis construction methods is that of reservoir sampling (Vitter 1985). • Great appeal because it generates a sample of the original multi-dimensional data representation. • Can be used with arbitrary data mining applications with little changes to the underlying algorithms.
Reservoir Sampling (Vitter 1985) • In the case of a fixed data set of known size N , it is trivial to construct a sample of size n , since all points have an inclusion probability of n/N . • However, a data stream is a continuous process, and it is not known in advance how many points may elapse before an analyst may need to use a representative sample. • The base data size N is not known in advance. • A reservoir or dynamic sample is maintained by probabilistic insertions and deletions on arrival of new stream points. • Challenge: Probabilistic insertions and deletions always need to maintain unbiased sample.
Reservoir Sampling • The first n points in the data stream are added to the reser- voir for initialization. • Subsequently, when the ( t +1)th point from the data stream is received, it is added to the reservoir with probability n/ ( t + 1). • This point replaces a randomly chosen point in the reservoir. • Note: Probability of insertion reduces with stream progres- sion. • Property: The reservoir sampling method maintains an un- biased sample of the history of the data stream (proof by induction).
Observations • In an evolving data stream only the more recent data may be relevant for many queries. • For example, if an application is queried for the statistics for the past hour of stream arrivals, then for a data stream which has been running over one year, only about 0 . 01% of an unbiased sample may be relevant. • The imposition of range selectivity or other constraints on the query will reduce the relevant estimated sample further. • In many cases, this may return a null or wildly inaccurate result.
Observations • In general, the quality of the result for the same query will only degrade with progression of the stream , as a smaller and smaller portion of the sample remains relevant with time. • This is also the most important case for stream analytics, since the same query over recent behavior may be repeatedly used with progression of the stream.
Potential Solutions • One solution is to use a sliding window approach for restrict- ing the horizon of the sample. • The use of a pure sliding window to pick a sample of the immediately preceding points may represent another extreme and rather unstable solution. • This is because one may not wish to completely lose the entire history of past stream data. • While analytical techniques such as query estimation may be performed more frequently for recent time horizons, distant historical behavior may also be queried periodically.
Biased Reservoir Sampling • A practical solution is to use a temporal bias function in order to regulate the choice of the stream sample. • Such a solution helps in cases where it is desirable to obtain both biased and unbiased results. • In some data mining applications, it may be desirable to bias the result to represent more recent behavior of the stream. • In other applications such as query estimation, while it may be desirable to obtain unbiased query results, it is more criti- cal to obtain accurate results for queries over recent horizons. • The biased sampling method allows us to achieve both goals.
Contributions • In general, it is non-trivial to extend reservoir maintenance algorithms to the biased case. In fact, it is an open problem to determine whether reservoir maintenance can be achieved in one-pass with arbitrary bias functions. • We theoretically show that in the case of an important class of memory-less bias functions (exponential bias functions), the reservoir maintenance algorithm reduces to a form which is simple to implement in a one-pass approach. • The inclusion of a bias function imposes a maximum require- ment on the sample size. Any sample satisfying the bias requirements will not have size larger than a function of N .
Contributions • This function of N defines a maximum requirement on the reservoir size which is significantly less than N . • In the case of the memory-less bias functions, we will show that this maximum sample size is independent of N and is therefore bounded above by a constant even for an infinitely long data stream. • We will theoretically analyze the accuracy of the approach on the problem of query estimation. • Test the method for the problem of query estimation and data mining problems.
Bias Function • The bias function associated with the r th data point at the time of arrival of the t th point ( r ≤ t ) is given by f ( r, t ). • The probability p ( r, t ) of the r th point belonging to the reser- voir at the time of arrival of the t th point is proportional to f ( r, t ). • The function f ( r, t ) is monotonically decreasing with t (for fixed r ) and monotonically increasing with r (for fixed t ). • Therefore, the use of a bias function ensures that recent points have higher probability of being represented in the sample reservoir.
Biased Sample • Definition: Let f ( r, t ) be the bias function for the r th point at the arrival of the t th point. A biased sample S ( t ) at the time of arrival of the t th point in the stream is defined as a sample such that the relative probability p ( r, t ) of the r th point belonging to the sample S ( t ) (of size n ) is proportional to f ( r, t ). • For the case of general functions f ( r, t ), it is an open problem to determine if maintenance algorithms can be implemented in one pass.
Challenges • In the case of unbiased maintenance algorithms, we only need to perform a single insertion and deletion operation periodi- cally on the reservoir. • In the case of arbitrary functions, the entire set of points within the current sample may need to re-distributed in order to reflect the changes in the function f ( r, t ) over different values of t . • For a sample S ( t ) this requires Ω( | S ( t ) | ) = Ω( n ) operations, for every point in the stream irrespective of whether or not insertions are made.
Memoryless Bias Functions • The exponential bias function is defined as follows: f ( r, t ) = e − λ ( t − r ) (1) • The parameter λ defines the bias rate and typically lies in the range [0 , 1] with very small values. • A choice of λ = 0 represents the unbiased case. The expo- nential bias function defines the class of memory-less func- tions in which the future probability of retaining a current point in the reservoir is independent of its past history or arrival time. • Memory-less bias functions are natural, and also allow for an extremely efficient extension of the reservoir sampling method.
Maximum Reservoir Requirements • Result: The maximum reservoir requirement R ( t ) for a ran- dom sample (without duplicates) from a stream of length t which satisfies the bias function f ( r, t ) is given by: t � R ( t ) ≤ f ( i, t ) /f ( t, t ) (2) i =1 • Proof Sketch: – Derive expression for probability p ( r, t ) in terms of reservoir size n and bias function f ( r, t ). t � p ( r, t ) = n · f ( r, t ) / ( f ( i, t )) (3) i =1 – Since p ( r, t ) is a probability, it is at most 1. – Set r = t to obtain result.
Maximum Reservoir Requirement for Exponential Bias Functions • The maximum reservoir requirement R ( t ) for a random sam- ple (without duplicates) from a stream of length t which satisfies the exponential bias function f ( r, t ) = e − λ ( t − r ) is given by: R ( t ) ≤ (1 − e − λt ) / (1 − e − λ ) (4) • Proof Sketch: Easy to show by instantiating result for gen- eral bias functions.
Constant Upper bound for Exponential Bias Functions • Result: The maximum reservoir requirement R ( t ) for a ran- dom sample from a stream of length t which satisfies the exponential bias function f ( r, t ) = e − λ ( t − r ) is bounded above by the constant 1 / (1 − e − λ ). • Approximation for small values of λ : The maximum reser- voir requirement R ( t ) for a random sample (without dupli- cates) from a stream of length t which satisfies the exponen- tial bias function f ( r, t ) = e − λ ( t − r ) is approximately bounded above by the constant 1 /λ .
Implications of Constant Upper Bound • For unbiased sampling, reservoir size may be as large as stream itself- no longer necessary for biased sampling! • The constant upper bound shows that maximum reservoir size is not sensitive to how long the points from the stream are being received. • Provides an estimate of the maximum sampling requirement. • We can maintain the maximum theoretical reservoir size if sufficient main memory is available.
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